#8696809

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#50011	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4471 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,5)(3,7)(9,10)(11,21)(12,18)(13,23)(14,19)(15,17)(16,20)(22,24)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(1,2,4,5)(3,6,7,8)(9,11,16,15)(10,17,20,21)(12,22,13,19)(14,23,24,18)(27,33)(30,36)(34,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,20,23)(11,22,15,19)(14,21,24,17)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,13,16,12)(10,18,20,23)(11,22,15,19)(14,17,24,21)(25,27,29,33)(26,30,32,36)(28,34,35,39)(31,37,38,40);(1,3,4,7)(2,6,5,8)(9,14,16,24)(10,19,20,22)(11,23,15,18)(12,21,13,17)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(9,15,16,11)(10,17,20,21)(12,19,13,22)(14,23,24,18)(25,28,29,35)(26,31,32,38)(27,34,33,39)(30,37,36,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8604005);